DFT Output is mirrored

[Sambit P]

I was trying to use the DFT plots from scipy, and I found that for a signal of length N, the DFT is an exact mirror after the N/2 point on the frequency plot. I cannot happen to understand why this happens.

[Sambit P]

Uploading plots of the signal I used and the DFT I got from it.

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[Catherine Lai]

This mirroring is a fundamental property of the DFT. We go through why it happens in the SIGNALS materials. i.e.:

https://github.com/laic/uoe_speech_processing_course/blob/master/signals/sp-m1-4-discrete-fourier-transform.ipynb

Try the exercise under ‘The DFT for k = 2 and beyond’ and see if you can see why this happens.

[Sambit P]

I understand that if the signal is a complex signal, this does not happen. The mirroring happens only if the signal has either a real or an imaginary component.
But I am finding it hard to understand the physical intuition of the mirroring.

[Simon]

We are usually only concerned with taking the DFT of real-valued signals, such as speech waveforms. So, we will always see this mirroring.

Here are some further clues:

1) the mirroring is centred around the Nyquist frequency
2) the signal does not contain any information above the Nyquist frequency
3) aliasing !

[Sambit P]

Ah, so as I understand, the first N/2 sample represent the real components (cosine values) and the last N/2 components represent the imaginary components (sine values). The mirroring happens because the cosine and sine values have a difference in phase of 90 degrees.

Is my understanding correct? Or have I missed something?

[Simon]

No, that’s not correct. The last N/2 values are identical to the first N/2 – they are just copies of the same values (their ordering is mirrored around the Nyquist frequency).

There are only N/2 magnitudes. No more.

The N numbers (samples) in the time domain (waveform) have been transformed into N/2 magnitudes and N/2 phases in the frequency domain. So, in the frequency domain (= magnitude spectrum & phase spectrum) there are also exactly N numbers.

In other words, the transform to the frequency domain has preserved all of the information in the waveform. That means the inverse transform will perfectly reconstruct the waveform.

[Sambit P]

Oh right. So this mainly happens because any frequency above the Nyquist frequency (NyF + F) behaves exactly like (NyF – F) and multiplying that with the signal basically gives the same value.
Makes sense. Thanks a lot!

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